Orthogonal representation of Weber's function using Hermite polynomials (Q5959027)

In this interesting paper, the author considers Weber's parabolic cylinder function \(D_\nu(z)\), which is defined in terms of hypergeometric functions, and admits the Maclaurin series expansion NEWLINE\[NEWLINED_\nu(z)= \exp \left(-{1 \over 4}z^2 \right)\sum^\infty_{j=0} {\Gamma(\frac 12) 2^{\nu-j \over 2}\over \Gamma\left( {1-\nu+j \over 2}\right)} {\nu\choose j}z^j.NEWLINE\]NEWLINE The author establishes an expansion for these in terms of Hermite polynomials: NEWLINE\[NEWLINED_\nu(z)= \exp\left(-{1\over 2}z^2 \right) \sum^\infty_{j=0} A_jH_j(x){_2F_1} \left({-j \over 2}, {1-j\over 2}; {v-j+2 \over 2}; -{1\over 16}\right).NEWLINE\]NEWLINE The \(\{A_j\}\) are explicitly given constants.